Scattering theory for wave equations with long-range perturbations
This work is devoted to the scattering theory for the wave equation with long-range perturbations: \[ Lu=\partial^ 2_ t u- \sum^{n}_{j,k=1}\partial_{x_ j} a^{jk}(x) \partial_{x_ k} u+V(x)u, \] where the symmetric matrix \(A(x)=(a^{jk}(x))\) is positively definite, V(x)\(\geq 0\) and \[ | \partial_ x^{\alpha}(a^{jk}(x)- \delta^{jk})| \leq C_{\alpha}(1+| x|)^{-| \alpha | -\delta},\quad 0<\delta <1;\quad | \partial_ x^{\alpha}V(x)| \leq C_{\alpha}(1+| x|)^{-| \alpha | -\delta} \] for all multi-indices \(\alpha\). The author proves the existence and completeness of the modified wave operators related to L and \(\partial^ 2_ t-\Delta\). For this purpose he establishes a general invariance principle involving as admissible functions real- valued \(C^{\infty}({\mathbb{R}}^+)\)-functions \(\phi\) (\(\lambda)\) with \(\phi '(\lambda)>0\) for all \(\lambda \in {\mathbb{R}}^+\).
scientific article; zbMATH DE number 3939867